Standard Deviation Calculator
Compute the mean, variance, and standard deviation of a data set (sample or population).
Inputs
Standard Deviation
13.4907
Mean
18.0000
Variance
182.0000
Count
6
Step by step
Collect the values
4, 8, 15, 16, 23, 42
= 6 values
Calculate the mean
(4 + 8 + 15 + 16 + 23 + 42) ÷ 6
= 18.000000
Compute each squared deviation (xᵢ − mean)²
(4.0000 − 18.0000)² = 196.000000; (8.0000 − 18.0000)² = 100.000000; (15.0000 − 18.0000)² = 9.000000; (16.0000 − 18.0000)² = 4.000000; (23.0000 − 18.0000)² = 25.000000; (42.0000 − 18.0000)² = 576.000000
Sum the squared deviations, then divide by n − 1 (5)
910.000000 ÷ 5
= 182.000000
Sample formula (Bessel's correction): divide by n − 1 to get an unbiased estimate.
Take the square root to get the standard deviation
√182.000000
= 13.4907
How it works
Standard deviation measures how spread out numbers are around their mean. Use the sample formula (dividing by n − 1) when your data is a sample of a larger group, and the population formula (dividing by n) when you have the entire population.
Formulas
Mean
Mean = (sum of all values) ÷ n
- x̄
- Arithmetic mean
- xᵢ
- Each individual value
- n
- Number of values
Sample variance (Bessel's correction)
Sample variance = Σ(xᵢ − mean)² ÷ (n − 1)
- s
- Sample standard deviation
- x̄
- Sample mean
- n
- Number of values
Population variance
Population variance = Σ(xᵢ − mean)² ÷ n
- σ
- Population standard deviation
- x̄
- Population mean
- n
- Total number of values
Standard deviation
Standard deviation = √variance
Frequently Asked Questions
Sample vs population — which do I pick?
If your numbers are a subset drawn from a bigger group, use Sample. If they represent every member of the group, use Population.
Why divide by n − 1 for a sample?
Dividing by n − 1 (Bessel's correction) removes bias when estimating the spread of the full population from a limited sample.